OBJ with Vertex Group export and import
Ok, I've just been testing myself, basically it's what I said before. OBJ file format specification probably doesn't support "vertex groups" feature.
There is no such structure or way to store that information in this file format.

What blender does apparently try to map vertex groups to the "poly groups" feature of OBJ. Since polygroups - as the name implies - only stores groups of faces it as a few caveats as it can't unequivocally map vertex groups to faces, so it can't store certain cases like single isolated vertex from vertex groups.

The minimum supported is probably three vertex, which is the least possible to define a face. This is what my four-corner vertex group (to the left) was imported like, two separate meshed with the four corners separated from the main one (colors for clarity purposes).



So short answer is it is not supported, either find another file format that supports this, recreate you vertex groups in your destination application, or fix them by hand there.